The calculation of effective properties of periodic suspensions is often problematic. Particularly difficult are calculations involving unit cells with many, close to touching, inclusions and high desired accuracy. In this paper we apply the conjugate gradient method and the fast multipole method to simplify calculations of this kind. We show how to dramatically speed up the computation of the effective conductivities and structural parameters for suspensions of disks and spheres. This enables accurate treatment of unit cells with thousands of inclusions. Direct estimates of the effective conductivity are compared with estimates via bounds. Accuracy of twelve digits is obtained for a suspension of disks which has been studied previously, but for which no accurate digit has been determined.